Engineering Structures 141 (2017) 482–494

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Engineering Structures

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Seismic collapse performance of special moment steel frames withtorsional irregularities

http://dx.doi.org/10.1016/j.engstruct.2017.03.0450141-0296/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: swhan82@gmail.com (S.W. Han).

Sang Whan Han ⇑, Tae-O Kim, Dong Hwi Kim, Seong-Jin BaekDepartment of Architectural Engineering, Hanyang University, Seoul 133-791, Republic of Korea

a r t i c l e i n f o

Article history:Received 24 February 2016Revised 16 March 2017Accepted 20 March 2017Available online 2 April 2017

Keywords:Torsional irregularityInelastic responseGround motionCollapse performanceStory driftNonlinear response history analysis

a b s t r a c t

Many building structures exhibit torsional irregularity, which is a type of horizontal irregularity. It is dif-ficult to estimate the inelastic response of torsionally irregular structures subjected to earthquake groundmotions using numerical analyses because torsionally irregular structures experience both lateral dis-placement and floor rotation. This study evaluates the collapse performance of multi-story model struc-tures with various degrees of torsional irregularity via nonlinear response history analyses. This studyalso proposes a procedure for computing design story drift demands, which allows torsionally irregularstructures to have uniform collapse risk irrespective of the degree of torsional irregularity.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Torsional irregularities are a type of horizontal irregularity(Table 12.3-1, ASCE 7-10 [1]) that are induced by an asymmetricdistribution of mass, stiffness and (or) strength. Since torsionallyirregular structures experience both lateral displacement and floorrotation during an earthquake, they sustain greater member forcesand drifts compared to regular structures. Without careful designconsideration, torsionally irregular structures may be more vulner-able to earthquakes than regular structures. To reduce the vulner-ability of torsionally irregular structures, current seismic codesspecify more stringent requirements for torsionally irregular struc-tures than regular structures; in other words, torsionally irregularstructures must be designed to satisfy the design requirements forregular structures as well as additional requirements for torsion-ally irregular structures.

Many studies have been conducted to investigate the effect oftorsional irregularity on the seismic response of single-story tor-sionally irregular structures designed according to seismic designprovisions (Chopra and Goel [2]; Tso and Wong [3]; Humar andKumar [4]; Dutta and Das [5]; Aziminejad and Moghadam [6]; Her-rera and Soberon [7]). Although it is convenient to draw insightabout the effect of torsional irregularity by analyzing simplesingle-story structures, there are limitations when it comes to

extrapolate the results from single-story structures to multi-storytorsionally irregular structures (Stathopoulos and Anagnos-topoulos [8]).

Recently, due to the development of high-performance comput-ers and software, the seismic behavior of multi-story structureswith torsional irregularity has been investigated by conductingnonlinear response analyses (Jeong and Elnashai [9]; Stathopoulosand Anagnostopoulos [10]; Reyes and Quintero [11]). De-la-Colina[12] investigated the seismic behavior of multi-story torsionallyirregular structures, and proposed design recommendations tocontrol the ductility demands. DeBock et al. [13] investigated theseismic collapse performance of multi-story reinforced concreteframe structures using sophisticated plastic models that accountedfor cyclic and in-cyclic deterioration in order to evaluate the designrecommendations related to accidental torsion.

The objective of this study is to evaluate the seismic collapseperformance of multi-story structures with various degrees of tor-sional irregularity, which are designed according to current seismicdesign provisions. For this purpose, three- and nine-story modelstructures with various degrees of torsional irregularity aredesigned according to ASCE 7-10 [1], AISC 341-10 [14] and AISC360-10 [15], in which special moment steel frames are used asseismic force resisting systems. Incremental dynamic analysesare conducted for the model structures with repeated nonlinearresponse history analyses using a three-dimensional inelastic ana-lytical model. Based on the analysis results, this study evaluatesthe collapse performance of the model structures, and proposes a

S.W. Han et al. / Engineering Structures 141 (2017) 482–494 483

procedure for computing the design story drift demand in anattempt to allow torsionally irregular structures to have uniformcollapse risk. In order to monitor the change in member sectionsof torsionally irregular structures according to the design stages,this study also estimates the total steel weight of the momentframes in model structures at each design stage.

2. Summary of design process for torsionally irregularstructures

In ASCE 7-10 (Table 12.3-1), when the maximum story drift atone end of the structure is more than 1.2 times the average storydrift at both ends of the structure, the structure is classified asbeing torsionally irregular. When the drift ratio is greater than1.4, the structure is considered to have extreme torsionalirregularity.

When a structure has torsional irregularity, the structure shouldbe designed to satisfy the design requirements for regular struc-tures as well as additional requirements for torsionally irregularstructures, as specified in ASCE 7-10 [1]. The seismic design pro-cess for torsionally irregular structures is similar to that of regularstructures. In this study, the design process for torsionally irregularstructures is separated into three stages: (1) strength design stage,(2) drift (stiffness) checking stage, and (3) stability (P � Deffect)checking stage. Fig. 1 illustrates the schematic flow of the seismicdesign process for torsionally irregular structures that is used inthis study.

The additional requirements specified in ASCE 7-10 are: (1)increases in forces due to irregularities for SDC D, E and

Determine seismic design forces

Calculate member force demands and select the member sections

Calculate story drift Δ(If torsional irregular, Δ shall be computed as the largest difference of the deflections)

OK

Calculate stability coefficient, θ

OK

Stage 1:Strength design

Stage 2:Drift check

Stage 3:Stability

check

OK

allowableΔ < Δ

maxθ θ<

0.1θ <

Fig. 1. Seismic design process for

F (12.3.3.4), (2) 3-D modeling for structures having torsional irreg-ularity type 1a, 1b, 4, or 5 of Table 12.3-1 in ASCE 7-10 (12.7.3), (3)amplification of accidental torsion for structures assigned to SDC Cthrough F (12.8.4.3), (6) analysis procedure selection for torsionallyirregular structures (12.6), and (7) story drift calculation using thelargest difference of the deflections of vertically aligned points atthe top and bottom of the story along any edges of the structure(12.8.6). The numbers in parentheses are the section numbers ofASCE 7-10 [1] for each of the additional requirements.

2.1. Strength design

In the strength design stage, seismic design forces are calcu-lated to determine the member section. Three different analysisprocedures are permitted in ASCE-7-10 [1]: (1) equivalent lateralforce analysis, (2) modal response spectrum analysis, and (3)response history analysis. A three-dimensional analytical modelis required for the analysis of torsionally irregular structuresassigned to all levels of SDCs, with the exception of SDC A. For tor-sionally irregular structures assigned to SDCs D, E, and F, the equiv-alent lateral force analysis procedure is not permitted. In thisstudy, a modal response spectrum analysis method that has nolimitations according to the level of SDCs is used.For torsionallyirregular structures, both inherent and accidental torsions shouldbe properly accounted for when calculating the member forcesand drifts. The inherent torsion results from an asymmetric distri-bution of mass, stiffness, and strength, whereas accidental torsionis induced by uncertainties in the distribution of mass, stiffness,and strength. The accidental torsional moment is calculated using

Modal spectral response analysis

NG

Change section

Amplify member force demandsand amplify story drift demands

Complete the design

If the structure is torsional irregular, amplify accidental torsional moment

Change section

Check the member sections

Detailing Requirements (AISC 341, 358)

OK

NG

NG

torsion-irregular structures.

484 S.W. Han et al. / Engineering Structures 141 (2017) 482–494

the assumed displacement of the center of mass in each directionfrom its actual location by a distance equal to 5% of the dimensionof the structure that is perpendicular to the direction of the appliedforces. The accidental torsional moment for torsion-irregular struc-tures assigned to SDCs C, D, E, and F should be amplified using atorsional amplification factor (Ax), which can be calculated usingEq. (1).

Ax ¼ dmax

1:2davg

� �2

ð1Þ

where dmax is the maximum displacement at level x, and davg is theaverage displacement at the extreme points of the structure at levelx (Fig. 2). Using proper analysis procedures, the member forces arecalculated by considering both the inherent and accidental torsion,and the adequate member sections are selected.

2.2. Drift check

After determining the proper member sections that are capableof resisting member forces, the design story drift of a structureshould be checked. According to Section 12.8.6 of ASCE 7-10, thedesign story drift (D) can be calculated as the difference of thedeflections (dCM) at the center of mass at the top and bottom ofthe story under consideration except for structures of SDCs C, D,E and F having horizontal irregularity Type 1a or 1b. For thesestructures, the design story drift is calculated as the largest differ-ence of deflections (dmax) along any of the edges of the structure attop and bottom of the story under consideration, which is the lar-gest difference in deflection observed at any column line. As shownin Fig. 2, dmax is larger than dCM . The design story drift (D) shouldnot exceed the allowable story drift that is specified inTable 12.12-1 of ASCE 7-10, which varies according to the structuretype and risk category. Otherwise, the process of selecting membersections is repeated until the drift requirement is satisfied.

2.3. Stability check

To incorporate the P � D effect, the stability coefficient (h) is cal-culated using Eq. (2):

h ¼ PxDIeVxhxCd

ð2Þ

where Px is the total vertical design load, Vx is the seismic shearforce in story x, hx is the story height, Ie is the importance factor,and Cd is the deflection amplification coefficient that is specified

BδAδ

CMδ

max

2 1.2A B

avg xavg

A δδ δδδ

⎡ ⎤+= = ⎢ ⎥⎢ ⎥⎣ ⎦

Fig. 2. Floor displacements in torsion-irregular structures.

in Table 12.2-1 of ASCE 7-10. The stability coefficient h should beless than hmax, as specified by Eq. (12.8-17) in ASCE 7-10 [1].

When h is greater than 0.1, P � D effects should be accounted forby increasing the member forces and the design story drifts basedon rational analyses. Alternatively, it is permitted to increase theseismic response demand by an amplification factor: 1=ð1� hÞ.In this study, the latter method is used to account for the P � Deffect.

Member sections, which were determined from previous designstages, should be checked to determine whether the members arecapable of sustaining the increased member forces and designstory drifts. The process of selecting member sections should berepeated until the structure is capable of resisting the increasedmember forces and drifts.

3. Seismic design of model structures

3.1. Model structures

Office buildings are designed according to ASCE7-10 [1], whichare considered as model structures. Fig. 3 shows the floor plans andelevations of the model structures. To sustain seismic loads, twosteel special moment frames (SMF) are placed in each orthogonaldirection as shown in Fig. 3b (marked by thicker solid lines). Forsteel SMFs, the response modification factor (R), deflection ampli-fication factor (Cd), overstrength factor (Xo), and importance factor(Ie) are 8, 5.5, 3, and 1, respectively. In Fig. 3b, the dotted linesdenote gravity frames. Among the configuration types of momentframes shown in Fig. 3b, Type 1 is the floor plan for regular struc-tures, in which moment frames are symmetrically placed along theperimeter of the structure.

In order to impart different degrees of torsional irregularity to astructure, the west side moment frame (W-frame) is placed at fivedifferent locations as shown in Fig 3b. As the W-frame approachesthe E-frame, the torsional irregularity becomes more severe. Thus,this study considers the six different floor plans shown in Fig. 3b. Itis noted that to place the W- frame inside a building rather thanalong the perimeter of the building is neither practical nor eco-nomical. Nevertheless, the different locations of the W-frame areconsidered because the objective of this study is to investigateeffect of different degrees of torsional irregularity on the seismicperformance of buildings.

As shown in Fig. 3b, the overall floor plan does not vary accord-ing to the location of the W-frame. Because heavy cast-in-placeconcrete rigid diaphragms are used in model structures, the totalweight of a building is much larger than that of the W-frame inthe building. The weight of the W-frame for three and nine storybuildings is less than 2.3% of the total weight of these building.Therefore, the center of mass was assumed to locate at the centerof plan irrespective of the location of the W-frame. Variability inmass center is not considered in the present study.

To investigate the effect of building height on the collapse per-formance of torsionally irregular structures, this study considersthree- and nine-story building structures. All building structureshave five bays in each direction (Fig. 3) where the story heightand span length are 4 m and 6 m, respectively. All structures aresymmetric about the east-west direction (x-axis), but are asym-metric about the north-south direction (y-axis), with the exceptionof Type 1 (regular) structures.The model structures are assumed tobe located at a site with design spectral response accelerations of1.0 g and 0.6 g at a short period (SDS) and a 1-s period (SD1), respec-tively; thus, according to ASCE 7-10 [1], the buildings are assignedto SDC D. The dead and live loads on the floors are 4.60 kPa and2.39 kPa, respectively, whereas the corresponding loads on the roofare 4.12 kPa and 0.96 kPa, respectively [16].

Fig. 3. Model structures.

S.W. Han et al. / Engineering Structures 141 (2017) 482–494 485

Member force and story drift demands are calculated using amodal response spectrumanalysis,which canbeused for torsionallyirregular structures regardless of SDC levels as described in ASCE7-10 [1]. The commercial software MIDAS-Gen [17] is used for theanalysis and design. Member sections are determined according toAISC 360-10 [15]; the strong column and weak beam requirementsspecified in E3 of AISC 341-10 [14] are also considered.

According to ASCE 7-10 [1], three- and nine-story buildingstructures with irregularity types 4, 5, and 6 (Fig. 3b) are classifiedas torsionally irregular structures. Alternatively, structures withirregularity types 1, 2, and 3 are classified as regular structures.Thus, torsionally irregular structures (3-TP4-O, 3-TP5-O, 3-TP6-O,9-TP4-O, 9-TP5-O, and 9-TP6-O) are designed according to therequirements for regular structures with additional requirementsfor torsionally irregular structures. It is noted that 3-TP4-O standsfor a three-story structure (3) of irregularity Type 4 (TP4) that isdesigned according to the requirements for torsionally irregularstructures (O). For nine-story structures [i.e., 9-TP5-O and9-TP6-O with severe torsion-irregularity (types 5 and 6)], member

sections cannot be found from the list of W-shaped sectionsprovided in AISC 360-10 [15]. This indicates that it may not be eco-nomical and feasible to construct buildings having nine stories orhigher if they have severe torsional irregularity. Thus, these struc-tures are excluded from the list of model structures. In Appendix B,the member sections for the 3-TP-5O building are summarized atindividual design stages.

For comparison purposes, this study also considers torsionallyirregular structures (3-TP4-X, 3-TP5-X, 3-TP6-X, 9-TP4-X, and3-TP4-X) having the samemember sections with regular structures(3-TP1 and 9-TP1), thus without taking into account the additionalrequirements for torsional irregular structures. Therefore, thesestructures do not satisfy the building code requirements, and theymay not be safe.

Tables 1 and 2 summarize member sections used for beams andcolumns in Type 1- and Type 5-three story model structures (3-TP1and 3-TP5-O), respectively. Each model structure is designedrepeatedly to find the optimal member sections consideringtorsion due to the change in the location of the W-frame.

Table 1Member sections for Type 1 three story building (3-TP1).

Story number (Ns) N and S frames E and W frames

Column Beam Column Beam

Exterior Interior Exterior Interior

1 W14x48 W14x68 W14x38 W14x74 W14x74 W14x482 W14x48 W14x68 W14x38 W14x74 W14x74 W14x483 W14x48 W14x38 W14x38 W14x53 W14x53 W12x45

Table 2Member sections for Type 5 three story building (3-TP5-O).

Ns N and S frames E frame W frame

Column Beam Column Beam Column Beam

Exterior Interior Exterior Interior Exterior Interior

1 W14x W24x W24x W14x W14x W14x W24x W24x W27x68 94 55 74 74 48 162 162 102

2 W14x W24x W24x W14x W14x W14x W24x W24x W27x68 94 55 74 74 48 162 162 94

3 W14x W14x W21x W14x W14x W12x W24x W24x W24x68 68 68 53 53 45 131 131 76

486 S.W. Han et al. / Engineering Structures 141 (2017) 482–494

It is observed that member sections in the W-frame of the tor-sionally irregular structure (3-TP5-O) are significantly larger thanthose of corresponding regular structure (3-TP1). With increasingthe degree of torsional irregularity from Type 1 to Type 5, membersections for N- and S-frames increase moderately. To resist theenlarged forces due to torsion, it is less effective to increase mem-ber sections of the E-frame compared to increasing member sec-tions in W-, N-, and S-frames. In this study, the same membersections are used for the E- frame of torsionally irregular structuresas those for the E-frame of corresponding regular structures.

Table 3 summarizes the natural periods of the first three modes.Centerline dimensions are used for calculating the natural periods.Gravity columns are not included in period calculation and all col-umns in the first story are rigidly fixed to the ground.

The present study investigates the deformed shapes of the firstthree modes of model structures and the mass participations oftwo translation components and one rotation component for thesemodes. The results are not included in this paper. For torsionallyregular structures (3TP-1 and 9TP-1), the first mode includes trans-lation in the y-direction exclusively, whereas the second modeincludes only translation in the x-direction. The third mode

Table 3Natural Periods of the model structures.

Number of stories Plan type ID

3 Type 1 3-TP1Type 2 3-TP2-X

3-TP2-OType 3 3-TP3-X

3-TP3-OType 4 3-TP4-X

3-TP4-OType 5 3-TP5-X

3-TP5-OType 6 3-TP6-X

3-TP6-O9 Type 1 9-TP1

Type 2 9-TP2-X9-TP2-O

Type 3 9-TP3-X9-TP3-O

Type 4 9-TP4-X9-TP4-O

includes only rotation. However, unlike the regular structures,the first and third modes of torsionally irregular structures containboth translational component along y-axis and rotation compo-nent about z-axis, whereas the 2nd mode of these structures con-tains only translation component along x-axis.

3.2. Total weight of moment frames at three different design stages

The total steel weight of the moment frames in the model struc-tures is estimated at three different seismic design stages: (Stage 1)strength design stage: member sections are determined to resistthe member force demands; (Stage 2) drift (stiffness) checkingstage: design story drift demands should be smaller than theallowable story drift, and member sections can be changed if nec-essary; and (Stage 3) stability checking stage: member forces anddesign story drifts are amplified to account for the P � D effects,and member sections are changed if necessary.Fig. 4a presentsthe total steel weight (W) of the moment frames in three-storymodel structures that are calculated at each design stage. At Stage1 (strength checking), the weight for all structures is nearly thesame regardless of the degree of torsional irregularity. For three-

Periods, T (s)

1st mode 2nd mode 3rd mode

1.39 1.27 0.951.41 1.27 0.871.40 1.27 0.871.50 1.27 0.891.44 1.22 0.861.65 1.27 0.871.14 1.09 1.051.83 1.27 0.821.19 1.17 1.122.34 1.28 0.891.09 0.79 0.342.61 2.51 1.562.71 2.63 1.732.62 2.49 1.732.92 2.63 1.772.73 2.45 1.993.26 2.64 1.712.40 2.32 1.71

0

500

1000

1500

2000

2500

1 2 3

Stee

l wei

ghts

(kN

)

Design stages

3-TP1

3-TP2-O

3-TP3-O

3-TP4-O

3-TP5-O

3-TP6-O

(a) 3-story

1000

1500

2000

2500

3000

3500

1 2 3

Stee

l wei

ghts

(kN

)

Design stages

9-TP1

9-TP2-O

9-TP3-O

9-TP4-O

(b) 9-story

Fig. 4. Total steel weights of the moment frames in structures for each design stage.

S.W. Han et al. / Engineering Structures 141 (2017) 482–494 487

story regular structures (3-TP1, 3-TP2-O, and 3-TP3-O), the weightis nearly constant irrespective of the design stages. However, fornine-story regular structures (9-TP1, 9-TP-2-O, and 9-TP-3-O),the weight at Stage 2 is significantly larger than the weight at Stage1; the weight at Stage 3 is also larger than the weight at Stage 2.This indicates that nine-story structures are affected by driftsand the P � D effect more than three-story structures.

For torsionally irregular structures (3-TP4-O, 3-TP5-O, 3-TP6-O,and 9-TP4-O), the weight increases sharply from Stage 1 to Stage 2.It is noted that when drifts are checked for torsionally irregularstructures, dmax is used instead of dCM , as shown in Fig. 2; thisrequires larger member sections. For three-story torsionally irreg-ular structures, the increase in the weight is small between Stage 2and Stage 3; alternatively, for nine-story torsionally irregularstructures, the increase in the weight is significant. This observa-tions confirms that nine-story torsionally irregular structures areinfluenced by the P � D effect more than three-story torsionallyirregular structures. At Stages 2 and 3, the weights for torsionallyirregular structures are much larger than those for correspondingregular structures. For example, the weights of 3-TP6-O and9-TP4-O are 3.9 and 1.5 times larger than those of 3-TP1 and9-TP1, respectively. As shown in Fig. 4, as the degree of torsionalirregularity increases, larger steel weights are required.

4. Nonlinear response history analyses of torsionally irregularstructures

4.1. Analytical model

Nonlinear response history analyses (RHA) with a three-dimensional analytical model are conducted to evaluate the seis-mic collapse performance of the model structures. For analyses,Opensees (Mazzoni et al. [18]) software is used. Fig. 5a presentsthe three-dimensional frame model used in the present study.Moment frames are denoted by thick solid lines. Gravity columnsare modeled as rigid elements with pins at their ends, whichaccount for the P � D effects due to gravity loads. Slabs are mod-eled to behave as rigid diaphragms by restraining in-plane slabdeformation.

Fig. 5b presents the analytical model for the beam-column con-nections in moment frames. Columns are modeled using fiber ele-ments (Fig. 5b-1), which effectively simulate interaction betweenaxial force and moment. The behavior of the fiber elements is rep-resented by a bilinear model with a post elastic slope of 3%. Fiveintegration points are used, equally spaced along the column(Fig. 5b). Neuenhofer and Filippou [19] reported that four to sixintegration points are typically sufficient to represent the spreadof plasticity along an element.

Although fiber section elements may not incorporate cyclicdeterioration in strength and stiffness, they can efficiently capturedistributed plasticity in members. Because most plastic hingesare developed in beams rather than columns in special momentframes owing to strong column-weak beam requirements, cyclicdeterioration may not be a serious problem for columns in SMFs.At each integration point, the member section is divided into 160fibers (64 fibers for each flange and 32 fibers for web). It is notedthat the 108 fibers offers an almost perfect match of the exactsolution [20].

Beams are modeled to behave in an elastic range (Fig. 5b-2).Inelastic beam behavior is represented by inelastic rotationalsprings placed at the ends of each beam (Fig. 5b-3), where My

and Mc are the yield and maximum moment strengths, respec-tively, while hp is the plastic rotation capacity, which can be calcu-lated using the equations proposed by Lignos and Krawinkler [21].Cyclic deterioration in strength and stiffness is not considered inthis study.

4.2. Nonlinear response history analysis (RHA) of torsionally irregularstructures

To investigate the effect of torsional irregularity on the seismicdemands of torsionally irregular structures, this study conductsnonlinear RHA under an orthogonal pair of horizontal groundmotions: these motions were recorded at the Beverly Hills stationwith an epicentral distance of 13 km during the Northridge earth-quake. Their peak ground accelerations (PGAs) are 0.42 g and0.52 g, respectively.

Fig. 6 shows the distribution of plastic hinges in the N- andS-frames in the three-story model structures (3-TP1, 3-TP5-X,and 3-TP5-O) that are obtained from nonlinear RHA. As mentionedearlier, 3-TP1 is a three-story regular structure, 3-TP5-X is atorsion-irregular structure with the same member sections as3-TP1, and 3-TP5-O is also a torsion-irregular structure that isdesigned according to the requirements for torsionally irregularstructures that are specified in ASCE 7-10 [1].

As shown in Fig. 6, the plastic hinges developed in the N- andS-frames of the three model structures are denoted by solid circles.For the regular structure (3-TP1), the distribution and number ofplastic hinges developed in the N- and S-frames are almost identi-cal; this result is expected. However, for 3-TP5-X, the distributionof plastic hinges in the N- and S-frames is significantly asymmetric(Fig. 6b). The numbers of plastic hinges in the N- and S-frames are36 and 10, respectively. Unlike 3-TP5-X, the N- and S-frames of3-TP5-O have an almost symmetric distribution of plastic hinges.The numbers of plastic hinges detected in the N- and S-frames of3-TP5-X are 31 and 32, respectively.

(b-3) Rotational spring

(b-2) Beam element

M

MMc

My

e p

(a) Frame model (b) Connection model

M

(b-1) Column element

Fig. 5. Analytical model.

(a) 3-TP1 (b) 3-TP5-X (c) 3-TP5-O

N-frame (35)

S-frame (36)

N-frame (36)

S-frame (10)

N-frame (31)

S-frame (32)

EW EWE

S

N

W0.52g

0.42g

S

N

S

N

*The number of plastic hinges

Fig. 6. Distribution of plastic hinges.

488 S.W. Han et al. / Engineering Structures 141 (2017) 482–494

To investigate the cause of the difference in plastic hinge distri-butions in the N- and S-frames, the displaced roof positions areplotted with thin solid lines at each time step of the nonlinearRHA, as shown in Fig. 7. The original roof position is also plottedwith a dotted line as a reference. In Fig. 7, the thick solid linesdenote the displaced roof positions when the N- and S-frame dis-placements are the largest.The ratios of the maximum displace-ment of the N-frame (Dmax�N) to the maximum displacement ofthe S-frame (Dmax�S) are 1.02, 2.22, and 1.3 for 3-TP1, 3-TP5-X,and 3-TP5-O, respectively. As expected, the ratio (Dmax�N/Dmax�S)for the regular structure is the smallest and the ratio for 3-TP5-Xis the largest.This study also evaluates the effect of torsional irreg-ularity on the collapse intensities. For this purpose, incrementaldynamic analyses (IDA) are conducted (Vamvatsikos and Cornell[22]; Moon et al. [23]). Collapse is defined as the state of globaldynamic instability in one or more stories of the structural system.

In this study, the collapse intensity is defined as a 5% dampedpseudo spectral acceleration at the fundamental period [SaðT1Þ],which is the ordinate of IDA curves. The approximate period(CuTa) is used as the fundamental period, which is calculatedaccording to ASCE 7-10 [1] (section 12.8.2); for the three-storystructures, CuTa is 0.74 s.

The approximate period of the three-story regular structure(3-TP1) is shorter than the fundamental period (=1.39 s) obtainedfrom eigenvalue analyses. For torsionally irregular structures hav-ing the same member sections, the period difference betweenapproximate and eigenvalue periods is more significant with anincrease in torsional irregularity. As shown in Table 3, these struc-tures have a longer period as the level of torsional irregularityincreases. However, for torsionally irregular structures properlydesigned according to seismic design provisions with additionalrequirements for torsionally irregular structures, the fundamental

(a) 3-TP1 (b) 3-TP5-X (c) 3-TP5-O

N-frame

S-frame

N-frameN-frame

S-frame S-frame

xy

Fig. 7. Displaced roof positions for three-story model structures.

S.W. Han et al. / Engineering Structures 141 (2017) 482–494 489

period obtained from eigenvalue analyses generally decreases asthe level of torsional irregularity. The eigenvalue periods of thesestructures are also shorter than that of the corresponding torsion-ally irregular structures having the same member sections as reg-ular structures. Therefore, the period difference for thesestructures is not as significant as torsional irregular structures hav-ing the same member sections as 3-TP1. Similar observation wasmade for 9-story torsional irregular structures.To determine thebi-directional acceleration spectrum, the acceleration responsesof an elastic single-degree-of-freedom system with a period ofCuTa are calculated subjected to individual ground motions. Thevector sum of the bi-directional acceleration responses is com-puted for each time step and the maximum is defined as the valueof the acceleration response spectrum [24]. The abscissa of the IDAcurve is represented by an engineering demand parameter (EDP).In this study, the maximum story drift ratio (hmax) is used as theEDP. In the IDA curve, the global dynamic instability is identifiedby a characteristic flattening of each IDA, referred to as the flatline,where the seismic demand increases greatly with only a slightincrease in the ground motion intensity. To conduct IDA, the samepair of ground motions recorded at the Beverly Hills station duringthe Northridge earthquake, is used (Fig. 8).

The collapse intensities of 3-TP1 and 3-TP5-X are 1.8 g and0.8 g, respectively. This indicates that the collapse intensities oftorsionally irregular structures with the same member sections

Fig. 8. IDA curves and c

as regular structures are significantly smaller than those of corre-sponding regular structures. The collapse intensity of 3-TP5-O is3.6 g, which is twice the collapse intensity of 3-TP1. The larger col-lapse intensity of 3-TP5-O is due to the heavy section sizes that areselected in accordance with the stringent requirements for torsion-ally irregular structures (ASCE 7-10).

5. Seismic collapse performance evaluation

This study evaluates the seismic collapse performance of tor-sionally irregular structures. Current seismic codes are intendedto provide buildings designed according to code requirements withthe capability to have a low probability of collapse if subjected tovery rare, intense ground motions (referred to as maximum con-sidered earthquake (MCE) ground motion) [1]. FEMA P-695 [25]provides a methodology for quantifying the system performanceand response parameters in seismic design. The two performanceobjectives used in FEMA P-695 [25] are: (1) The probability of col-lapse for maximum considered earthquake (MCE) ground motionsis 10% or less, on average, across a performance group containingmodel frames with a specific seismic force resisting system withdifferent configurations, and (2) the probability of collapse is 20%or less for an individual model frame.The probability of collapseunder MCE ground motions [PðCollapsejSMTÞ] is calculated usingEq. (3).

ollapse intensities.

490 S.W. Han et al. / Engineering Structures 141 (2017) 482–494

PðCollapsejSMTÞ ¼ UlnðSMTÞ � lnðSCT � SSFÞ

bTOT

!ð3Þ

where UðxÞ is the cumulative distribution function of the standardnormal variate x;SMT is the MCE 5% damped-spectral response accel-

eration at the fundamental period (T) of the structure, SCT is themedian value of the 5% damped spectral response acceleration atT for a collapse level earthquake, SSF is the spectral shape factoraccounting for the spectral shape of rare groundmotions, and adjustthe collapse probability, UðxÞ, while bTOT is the total system collapseuncertainty calculated using Eq. (4).

bTOT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2RTR þ b2

DR þ b2TD þ b2

MDL

qð4Þ

where bRTR; bDR; bTD and bMDL are the record-to-record uncertainty,design requirement-related uncertainty, test data-related uncer-tainty and modeling uncertainty, respectively. These values rangefrom 0.1 to 0.5, and are determined from tables in pages 3–8, 3–20, and 5–23 of FEMA P-695 [25]. The step-by-step procedure usedto calculate the probability of collapse is described in Appendix A.

The three- and nine-story structures are assigned to SDC D. Forthese structures, SMT values are 1.15 g and 0.51 g, respectively.Table 4 summarizes the probabilities of collapse for modelstructures, which are calculated using the procedure listed inAppendix A.

Fig. 9 presents the effect of the degree of torsional irregularityon the probability of collapse in which the abscissa is the degreeof torsional irregularity (Fig. 3) and the ordinate is the probabilityof collapse. Twenty-two pairs of ground motions, provided inFEMA P695 [25], are used as input ground motions. Fig. 9 includesthe probabilities of collapse for torsionally irregular structuresdesigned with and without the additional requirements for tor-sional irregularity.

For three-story torsionally irregular structures with the samemember sections as those of regular structures, the probability ofcollapse becomes larger as the degree of torsional irregularityincreases. The collapse probabilities of 3-TP2-X and 3-TP3-X arehigher than 0.1, which is a limiting value for performance group,whereas exceed those of 3-TP4-X, 3-TP5-X, and 3-TP6-X exceed0.2, which is the limiting value for individual structures [25].

In contrast, for three-story torsionally irregular structuresdesigned with the additional requirements, the probability of col-lapse becomes smaller as the degree of torsional irregularity

Table 4Probability of collapse for the model structures.

Number of stories Model structures SMT

3 3-TP1 1.143-TP2-X 1.143-TP3-X 1.143-TP4-X 1.143-TP5-X 1.143-TP6-X 1.143-TP2-O 1.143-TP3-O 1.143-TP4-O 1.143-TP5-O 1.143-TP6-O 1.143-TP4-OM 1.143-TP5-OM 1.143-TP6-OM 1.14

9 9-TP3-X 0.519-TP4-X 0.519-TP2-O 0.519-TP3-O 0.519-TP4-O 0.519-TP4-OM 0.51

increases. The collapse probabilities of 3-TP2-O and 3-TP3-O areslightly higher than 0.1 (<0.2), whereas those of 3-TP4-O, 3-TP5-O,and 3-TP6-O (=0.0441, 0.0123, and 0.0062) aremuch lower than 0.1.

For nine-story torsionally irregular structures, similar observa-tions are also made. For torsionally irregular structures with thesame member sections as those of the regular structures, the prob-ability of collapse becomes higher as the degree of torsional irreg-ularity increases; however, this increment is not as significant asthat of the three-story torsionally irregular structures. For thenine-story torsionally irregular structures that are designedaccording to the additional requirements for torsion, the probabil-ity of collapse becomes smaller as the degree of irregularityincreases. It is noted that 9-TP4-O exhibited the smallest collapseprobability (=0.0752). Among eight torsionally irregular modelstructures designed according to additional requirements for tor-sion specified in ASCE 7-10, four structures have the probabilityof collapse lower than 0.1, which are 3-TP4-O, 3-TP5-O, 3-TP6-O,and 9-TP4-O.

Therefore, it is observed that as the degree of torsional irregu-larity increases, the collapse probability of torsionally irregularstructures that are designed according to the additional require-ments becomes smaller. This phenomenon is attributed to theheavy member sections selected for torsionally irregular struc-tures, which serve to fulfill the drift requirement.

6. Proposed procedure for computing the design story drift fortorsionally irregular structures

As shown in Fig. 9, the probability of collapse for code-compliant torsionally irregular structures is much smaller thanthat of regular structures. The probability of collapse also decreasesas the degree of torsional irregularity increases. This is attributedto the large member sections of the torsionally irregular structures.

In order to design torsionally irregular structures that have asimilar probability of collapse to regular structures, irrespectiveof the degree of torsional irregularity, this study proposes a proce-dure to determine the design story drift demands that are requiredin the drift checking stage (Fig. 2). As shown in Fig. 4, the sharpestincrease in the total steel weight was observed at the drift checkingstage.

In the proposed procedure, the drift demand of torsionallyirregular structures is estimated by calculating the differencebetween the displacements at the mass centers of two adjacent

lT SCT SSF PðCollapsejSMT Þ

5.64 1.70 1.33 0.125.75 1.65 1.33 0.125.14 1.61 1.31 0.144.57 1.30 1.29 0.223.75 1.30 1.26 0.239.44 0.85 1.14 0.365.63 1.65 1.34 0.126.49 1.70 1.36 0.116.60 2.40 1.36 0.046.15 3.60 1.35 0.015.36 4.38 1.32 0.018.49 1.70 1.42 0.142.77 1.90 1.21 0.134.39 2.86 1.29 0.063.14 0.59 1.23 0.233.82 0.55 1.26 0.253.99 0.64 1.27 0.194.09 0.70 1.27 0.154.61 0.93 1.29 0.084.59 0.82 1.29 0.10

Fig. 9. Collapse probabilities of the model structures.

S.W. Han et al. / Engineering Structures 141 (2017) 482–494 491

floors, as opposed to calculating the maximum drift values of thevertical members in the story, which is the same procedure for reg-ular structures.

Four torsionally irregular structures (3-TP4-OM, 3-TP5-OM 3, 3-TP6-OM, and 9-TP4-OM) are re-designed with the proposed proce-dure for drift calculations. Nonlinear response history analyses areconducted for structures that are subjected to the same pair ofground motions that were used in the previous section. The prob-ability of collapse for these structures calculated according toFEMA P-695 is summarized in Table 4.

Fig. 10 shows the distribution of plastic hinges in the N- and S-frames of the newly designed three-story structure (3-TP5-OM)with torsional irregularity Type 5. The numbers of plastic hingesin the N-and S-frames of 3-TP5-OM are 30 and 22, respectively(Fig. 10b). Thus, the difference between the numbers of plastichinges in the N-and S-frames of 3-TP5-OM is 12. It is noted that,for 3-TP5-X and 3-TP-5-O, the differences are 26 and 1, respec-tively (Fig. 6). This indicates that the contribution of torsional

Fig. 10. Results of nonline

responses in 3-TP5-OM is more significant than that in 3-TP5-O,but less than that in 3-TP5-X.

IDA is also conducted for 3-TP5-OM under the same pair ofground motions (Fig. 10c). The collapse intensity is 1.9 g, whichis less than that of 3-TP5-O (=3.6 g), but close to that of the regularframe 3-TP1 (=1.7 g).

This study also calculated the probabilities of collapse for 3-TP4-OM, 3-TP5-OM, 3-TP6-OM and 9-TP4-OM using the 22 pairsof ground motions provided in FEMA P695 [25], which were alsoused previously.

Fig. 11 shows the probability of collapse. In this figure, ‘ASCE 7’indicates the torsionally irregular structures designed according toASCE 7-10, whereas ‘MC’ denotes the torsionally irregular struc-tures designed according to ASCE 7-10 [1] with drift demands cal-culated using the proposed procedure. The collapse probability of‘MC’ structures is more uniform than that of the ‘ASCE 7’ structures.Also, the collapse probability of ‘MC’ structures is larger than that ofthe ‘ASCE 7’ structures but is similar with that of regular structures.

ar RHA for 3-TP5-OM.

Fig. 11. Probability of collapse according to the degree of torsional irregularity.

Fig. 12. Total steel weights required for the model structures.

492 S.W. Han et al. / Engineering Structures 141 (2017) 482–494

The collapse probabilities of three-story ‘MC’ structures (3-TP4-OM, 3-TP5-OM, and 3-TP6-OM) with different degrees of torsionalirregularity are 0.1372, 0.1307, and 0.0549, respectively. Alterna-tively, the collapse probabilities of three-story ‘ASCE 70 structures(3-TP4-O, 3-TP5-O, and 3-TP6-O) are 0.0441, 0.0123, and 0.0062,respectively. This indicates that ‘MC’ structures have probabilitiesof collapse that are closer to those of regular structures than ‘ASCE70 structures. It is also noted that ‘MC’ structures produced moreuniform probabilities of collapse according to the degree of tor-sional irregularity than ‘ASCE 70 structures. Similar observationsare also made for the nine-story structures (Fig. 11b). This studyalso compared the total steel weight of the structures as shownin Fig. 12. The total steel weight of ‘MC’ structures, using the pro-posed design methods is less than that of ‘ASCE7’ structures. It isnoted that there may be other design solutions that would bringthe ‘MC’ buildings to the desired collapse level.

7. Conclusions

In order to evaluate the seismic collapse performance of tor-sionally irregular structures, three- and nine-story steel specialmoment frames in model structures with various degrees of tor-sional irregularity were designed according to ASCE 7-10 [1]. Non-linear response history analyses were conducted. The conclusionsand recommendations of this study are as follows:

(1) When the degree of torsional irregularity was increased, lar-ger member sections were required. At the strength design

stage (Stage 1), the total steel weight of the moment framesin torsionally irregular structures was similar to that of reg-ular structures. However, at the drift checking stage (Stage2), the weight of torsionally irregular structures was muchlarger than that of regular structures.

(2) The increase in the total steel weight for three-story torsion-ally irregular structures was negligible between Stage 2 andStage 3; however, the increase for nine- story torsionallyirregular structures was significant between Stage 2 andStage 3 due to the large P � D effects.

(3) It is observed that for regular structures, the distribution ofplastic hinges was almost symmetric in the N- and S-frames. Torsionally irregular structures designed accordingto ASCE 7-10 [1] also exhibited a symmetric distribution ofplastic hinges in the N- and S-frames; this was due to thelarge member sections that were selected to satisfy the addi-tional requirements for torsionally irregular structures, asspecified in ASCE 7-10 [1].

(4) For torsionally irregular structures designed without consid-ering the additional requirements, the probability of collapseis larger as the degree of torsional irregularity increases.However, torsionally irregular structures that were designedwhile also considering the additional requirements gener-ally exhibited smaller probabilities of collapse compared toregular structures.

(5) The probabilities of collapse of torsionally irregular struc-tures, designed with drift demands calculated using the pro-posed method, is similar to those of regular structuresregardless of the degree of torsional irregularity.

S.W. Han et al. / Engineering Structures 141 (2017) 482–494 493

(6) The results of this study is obtained using only torsionallyirregular structures with steel SMF frames. Therefore, cau-tion should be made for buildings having different seismicforce resisting systems such as braced frames and shearwalls.

Acknowledgements

Authors would like to acknowledge the financial supports pro-vided by the National Research Foundation of Korea (No.2014R1A2A1A11049488). The valuable comments offered byanonymous reviewers are greatly appreciated.

Appendix A

A step-by-step procedure for calculating the probability of col-lapse according to FEMA P-695 [23]:

(1) Idealize the model frame using a proper analytical model.(2) Conduct nonlinear static analysis to determine the over-

strength factor (X) and period-based ductility (lT). This isdone by using Eqs. (A1) and (A2) with Eq. (A3), respectively[Fig. A1(a)].

X ¼ Vmax

VdðA1Þ

lT ¼ dudyeff

ðA2Þ

dy;eff ¼ C0Vmax

Wg

4p2

h iðmaxðTn; T1ÞÞ2 ðA3Þ

Fig. A1. Procedure for calculating the SSF and calcul

where Vmax is the maximum base shear resistance, Vd is thedesign base shear, du is the roof drift displacement at thepoint of 20% strength loss (=0.8 Vmax), dyeff is the effectiveyield roof displacement, Co relates the displacement of thesingle-degree-of-freedom system at the fundamental modeof a frame to the roof displacement of the frame, g is the grav-ity constant, Tn is the fundamental period (defined as CuTa,specified in ASCE 7-10) [1], and T1 is the fundamental periodof the model frame computed via eigenvalue analysis.

(3) Calculate the spectral shape factor (SSF),that accounts forthat accounts for the spectral shape of rare ground motionsas depicted in Fig. A1(c).

SSF ¼ exp½b1ð�eoðTÞ � �eðTÞrecord� ðA4Þ

�eoðTÞrecord ¼ 0:6ð1:5� TÞ ðA5Þ

b1 ¼ 0:14ðlT � 1Þ0:42 ðA6Þ

(4) Calibrate the probability of collapse by applying the SSF.

PðCollapsejSMTÞ ¼ UlnðSMTÞ � lnðSCT � SSFÞ

bTOT

!ðA7Þ

Appendix B

The following table includes information about member sec-tions for the Type 5-three story structure at three different designstages.

Tables B-1, B-2 and B-3.

ating the probability of collapse.

Table B-1Member sections used for the Type 5-three story structure at strength design stage.

Level N-and S-frames E-frame W-frame

Colum Girder Colum Girder Colum Girder

Exterior Interior Exterior Interior Exterior Interior

1 W14x48 W14x68 W14x38 W14x W14x W14x W14x W14x W14x74 74 48 74 74 48

2 W14x48 W14x68 W14x38 W14x W14x W14x W14x W14x W14x74 74 48 74 74 48

3 W14x38 W14x38 W12x35 W14x W14x W12x W14x W14x W12x53 53 45 68 68 45

Table B-2Member sections used for the Type 5-three story structure at drift checking stage.

Level N-and S-frames E-frame W-frame

Colum Girder Colum Girder Colum Girder

Exterior Interior Exterior Interior Exterior Interior

1 W14x68 W24x94 W24x55 W14x W14x W14x W24x W24x W27x74 74 48 162 162 102

2 W14x68 W24x94 W24x55 W14x W14x W14x W24x W24x W27x74 74 48 162 162 94

3 W14x68 W14x68 W21x68 W14x W14x W12x W24x W24x W24x53 53 45 131 131 76

Table B-3Member sections used for the Type 5-three story structure at stability checking stage.

Level N-and S-frames E-frame W-frame

Colum Girder Colum Girder Colum Girder

Exterior Interior Exterior Interior Exterior Interior

1 W14x68 W24x94 W24x55 W14x W14x W14x W24x W24x W27x74 74 48 162 162 102

2 W14x68 W24x94 W24x55 W14x W14x W14x W24x W24x W27x74 74 48 162 162 94

3 W14x68 W14x68 W21x68 W14x W14x W12x W24x W24x W24x53 53 45 131 131 76

494 S.W. Han et al. / Engineering Structures 141 (2017) 482–494

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